\section{向量组的线性相关和线性无关}

	\begin{titwo}
		$n$ 维向量组 $\bm \alpha_{1}, \bm \alpha_{2}, \cdots, \bm \alpha_{s} (3 \leq s \leq n)$ 线性无关的充要条件是\kuo.

		\onech{存在一组全为零的数 $k_{1},k_{2},\cdots,k_{s}$，使 $k_{1} \bm \alpha_{1} + k_{2} \bm \alpha_{2} + \cdots + k_{s} \bm \alpha_{s} = \bm 0$}{$\bm \alpha_{1}, \bm \alpha_{2}, \cdots, \bm \alpha_{s}$ 中任意两个向量都线性无关}{$\bm \alpha_{1}, \bm \alpha_{2}, \cdots, \bm \alpha_{s}$ 中任意一个向量都不能由其余向量线性表出}{存在一组不全为零的数 $k_{1},k_{2},\cdots,k_{s}$，使 $k_{1} \bm \alpha_{1} + k_{2} \bm \alpha_{2} + \cdots + k_{s} \bm \alpha_{s} \ne \bm 0$}
	\end{titwo}

	\begin{titwo}
		已知向量组 $\bm \alpha_{1} , \bm \alpha_{2} , \bm \alpha_{3} , \bm \alpha_{4}$ 线性无关，则向量组 $2 \bm \alpha_{1} + \bm \alpha_{3} + \bm \alpha_{4}, \bm \alpha_{2} - \bm \alpha_{4}, \bm \alpha_{3} + \bm \alpha_{4}, \bm \alpha_{2} + \bm \alpha_{3}, 2 \bm \alpha_{1} + \bm \alpha_{2} + \bm \alpha_{3}$ 的秩是\kuo.

		\fourch{$1$}{$2$}{$3$}{$4$}
	\end{titwo}

	\begin{titwo}
		已知 $3$ 维向量组 $\bm \alpha_{1} , \bm \alpha_{2} , \bm \alpha_{3}$ 线性无关，则向量组 $\bm \alpha_{1} - \bm \alpha_{2}, \bm \alpha_{2} - k \bm \alpha_{3}, \bm \alpha_{3} - \bm \alpha_{1}$ 也线性无关的充要条件是\htwo.
	\end{titwo}

	\begin{titwo}
		设有两个 $n$ 维向量组
		\begin{align*}
			(\text{\Rmnum{1}})&\bm \alpha_{1}, \bm \alpha_{2}, \cdots, \bm \alpha_{s},\\
			(\text{\Rmnum{2}})&\bm \beta_{1}, \bm \beta_{2}, \cdots, \bm \beta_{s},
		\end{align*}
		若存在两组不全为零的数 $k_{1},k_{2},\cdots,k_{s},\lambda_{1},\lambda_{2},\cdots,\lambda_{s}$，使 $(k_{1} + \lambda_{1}) \bm \alpha_{1} + (k_{2} + \lambda_{2}) \bm \alpha_{2} + \cdots + (k_{s} + \lambda_{s}) \bm \alpha_{s} + (k_{1} - \lambda_{1}) \bm \beta_{1} + \cdots + (k_{s} - \lambda_{s}) \bm \beta_{s} = \bm 0$，则\kuo.

		\onech{$\bm \alpha_{1} + \bm \beta_{1}, \cdots, \bm \alpha_{s} + \bm \beta_{s}, \bm \alpha_{1} - \bm \beta_{1}, \cdots, \bm \alpha_{s} - \bm \beta_{s}$ 线性相关}{$\bm \alpha_{1} + \bm \beta_{1}, \cdots, \bm \alpha_{s} + \bm \beta_{s}, \bm \alpha_{1} - \bm \beta_{1}, \cdots, \bm \alpha_{s} - \bm \beta_{s}$ 线性无关}{$\bm \alpha_{1}, \cdots, \bm \alpha_{s}$ 及 $\bm \beta_{1}, \cdots, \bm \beta_{s}$ 均线性相关}{$\bm \alpha_{1}, \cdots, \bm \alpha_{s}$ 及 $\bm \beta_{1}, \cdots, \bm \beta_{s}$ 均线性无关}
	\end{titwo}

	\begin{titwo}
		已知 $n$ 维向量组 $\bm \alpha_{1}, \bm \alpha_{2}, \cdots, \bm \alpha_{s}$ 线性无关，则向量组 $\bm \alpha_{1}', \bm \alpha_{2}', \cdots, \bm \alpha_{s}'$ 可能线性相关的是\kuo.

		\onech{$\bm \alpha_{i}'(i = 1,2,\cdots,s)$ 是 $\bm \alpha_{i}(i = 1,2,\cdots,s)$ 中第一个分量加到第 $2$ 个分量得到的向量}{$\bm \alpha_{i}'(i = 1,2,\cdots,s)$ 是 $\bm \alpha_{i}(i = 1,2,\cdots,s)$ 中第一个分量改变成其相反数的向量}{$\bm \alpha_{i}'(i = 1,2,\cdots,s)$ 是 $\bm \alpha_{i}(i = 1,2,\cdots,s)$ 中第一个分量改为 $0$ 的向量}{$\bm \alpha_{i}'(i = 1,2,\cdots,s)$ 是 $\bm \alpha_{i}(i = 1,2,\cdots,s)$ 中第 $n$ 个分量后再增添一个分量的向量}
	\end{titwo}

	\begin{titwo}
		设向量组 $\bm \alpha_{1}, \bm \alpha_{2}, \cdots, \bm \alpha_{s}(s \geq 2)$ 线性无关，且
		\begin{gather*}
			\bm \beta_{1} = \bm \alpha_{1} + \bm \alpha_{2},
			\bm \beta_{2} = \bm \alpha_{2} + \bm \alpha_{3},\\
			\cdots,\\
			\bm \beta_{s - 1} = \bm \alpha_{s - 1} + \bm \alpha_{s},
			\bm \beta_{s} = \bm \alpha_{s} + \bm \alpha_{1}.
		\end{gather*}
		讨论向量组 $\bm \beta_{1}, \bm \beta_{2}, \cdots, \bm \beta_{s}$ 的线性相关性.
	\end{titwo}

	\begin{titwo}
        已知向量组 $\bm \alpha_{1}, \bm \alpha_{2}, \cdots, \bm \alpha_{s + 1}(s > 1)$ 线性无关，
        \[
            \bm \beta_{i} = \bm \alpha_{i} + t \bm \alpha_{i+1}, i = 1,2,\cdots,s.
        \]
        证明：向量组 $\bm \beta_{1},\bm \beta_{2},\cdots,\bm \beta_{s}$ 线性无关.
	\end{titwo}

	\begin{titwo}
		设 $\bm A$ 是 $3 \times 3$ 矩阵，$\bm \alpha_{1},\bm \alpha_{2},\bm \alpha_{3}$ 是 $3$ 维列向量，且线性无关，已知
		\[
			\bm A \bm \alpha_{1} = \bm \alpha_{2} + \bm \alpha_{3},
			\bm A \bm \alpha_{2} = \bm \alpha_{1} + \bm \alpha_{3},
			\bm A \bm \alpha_{3} = \bm \alpha_{1} + \bm \alpha_{2}.
		\]
		\begin{enumerate}
			\item 证明 $\bm A \bm \alpha_{1},\bm A \bm \alpha_{2},\bm A \bm \alpha_{3}$ 线性无关;
			\item 求 $|\bm A|$.
		\end{enumerate}
	\end{titwo}

	\begin{titwo}
		已知 $\bm A$ 是 $n$ 阶矩阵，$\bm \alpha_{1}, \bm \alpha_{2}, \cdots, \bm \alpha_{s}$ 是 $n$ 维线性无关向量组，若 $\bm A \bm \alpha_{1}, \bm A \bm \alpha_{2}, \cdots, \bm A \bm \alpha_{s}$ 线性相关. 证明：$\bm A$ 不可逆.
	\end{titwo}

	\begin{titwo}
		设 $\bm A$ 是 $n \times m$ 矩阵，$\bm B$ 是 $m \times n$ 矩阵，$\bm E$ 是 $n$ 阶单位矩阵. 若 $\bm A \bm B = \bm E$，证明：$\bm B$ 的列向量组线性无关.
	\end{titwo}

	\begin{titwo}
		设 $\bm A$ 为 $n$ 阶正定矩阵，$\bm \alpha_{1}, \bm \alpha_{2}, \cdots, \bm \alpha_{n}$ 为 $n$ 维非零列向量，且满足
		\[
			\bm \alpha_{i}^{\TT} \bm A^{-1} \bm \alpha_{j} = 0(i \ne j; i,j = 1,2,\cdots,n).
		\]
		试证：向量组 $\bm \alpha_{1}, \bm \alpha_{2}, \cdots, \bm \alpha_{n}$ 线性无关.
	\end{titwo}

	\begin{titwo}
		设 $\bm A, \bm B, \bm C$ 均是 $3$ 阶矩阵，满足 $\bm A \bm B = - 2 \bm B, \bm C \bm A^{\TT} = 2 \bm C$. 其中
		\[
			\bm B = \begin{bsmallmatrix}
				1 & 2 & 3 \\
				-1 & 1 & 0 \\
				2 & -1 & 1
			\end{bsmallmatrix},
			\bm C = \begin{bsmallmatrix}
				1 & -2 & 1 \\
				-2 & 4 & -2 \\
				-1 & 2 & -1
			\end{bsmallmatrix}.
		\]
		\begin{enumerate}
			\item 求 $\bm A$;
			\item 证明：对任何 $3$ 维向量 $\bm \xi$，$\bm A^{100} \bm \xi$ 与 $\bm \xi$ 必线性相关.
		\end{enumerate}
	\end{titwo}